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|{{EquationRef|2}}}}

|{{EquationRef|2}}}}

Here, [math]\mathcal{J}[/math] is the dimension-averaged <math>EI</math> (see the entry effective information), <math>\mathrm{\phi}</math> is the coarse-graining strategy function, <math>f_{q}</math> is the macroscopic dynamics, <math>q</math> is the dimension of the coarsened macroscopic state, [math]\hat{X}_{t + 1}[/math] is the prediction of the microscopic state at time <math>t + 1</math> by the entire framework. This prediction is obtained by performing inverse coarse-graining operation (the inverse coarse-graining function is [math]\phi^{\dagger}[/math]) on the macroscopic state prediction <math>\hat{Y}_{t + 1}</math> at time <math>t + 1</math>. Here <math>\hat{Y}_{t + 1}\equiv f_q(Y_t)[/math] is the prediction of the macroscopic state at time <math>t + 1</math by the dynamics learner according to the macroscopic state <math>Y_t[/math] at time <math>t</math>, where <math>Y_t\equiv \phi(X_t)[/math] is the macroscopic state at time <math>t</math>, which is obtained by coarse-graining <math>X_t[/math] by <math>\phi[/math]. Finally, the difference between <math>\hat{X}_{t + 1}</math> and the real microscopic state data <math>X_{t + 1}</math> is compared to obtain the microscopic prediction error.

Here, [math]\mathcal{J}[/math] is the dimension-averaged <math>EI</math> (see the entry effective information), <math>\mathrm{\phi}</math> is the coarse-graining strategy function, <math>f_{q}</math> is the macroscopic dynamics, <math>q</math> is the dimension of the coarsened macroscopic state, [math]\hat{X}_{t + 1}[/math] is the prediction of the microscopic state at time <math>t + 1</math> by the entire framework. This prediction is obtained by performing inverse coarse-graining operation (the inverse coarse-graining function is [math]\phi^{\dagger}[/math]) on the macroscopic state prediction <math>\hat{Y}_{t + 1}</math> at time <math>t + 1</math>. Here [math]\hat{Y}_{t + 1}\equiv f_q(Y_t)[/math] is the prediction of the macroscopic state at time <math>t + 1</math> by the dynamics learner according to the macroscopic state [math]Y_t[/math] at time <math>t</math>, where [math]Y_t\equiv \phi(X_t)[/math] is the macroscopic state at time <math>t</math>, which is obtained by coarse-graining [math]X_t[/math] by [math]\phi[/math]. Finally, the difference between [math]\hat{X}_{t + 1}[/math] and the real microscopic state data [math]X_{t + 1}[/math] is compared to obtain the microscopic prediction error.

The entire optimization framework is shown below:

The entire optimization framework is shown below: